1. society
2. work
3. unions

Why factor income shares seem to be constant?
Hernando Zuleta
ITAM
Río Hondo 1, San Angel
México D.F., México
e-mail: [email protected]
December 2003
Abstract
The common assumptions that labor income share does not change over time and that factor income shares are
equal to the elasticity of output with respect to factors have had important implications for economic theory.
However, there are various theoretical reasons why the elasticity of output with respect to reproducible factors
should be correlated with the stage of development. In particular, the behavior of international trade and capital
flows and the existence of factor saving innovations imply such a correlation. If this correlation exists and if
factor income shares are equal to the elasticity of output with respect to factor, the labor income share must be
negatively correlated with the stage of development. We propose two complementary explanations for why
labor income share has no trend: (i) the existence of a labor intensive sector which produces non tradable goods;
and (ii) the existence of more than two factors. It is possible that an increasing trend in physical capital income
share is compensated by a decreasing trend in land income shares. Similarly, an increasing trend in human
capital income share may be compensated by a decreasing trend in raw labor income share.
JEL classification: E1, F0, O0.
Keywords: Factor Income Shares, Elasticity of output with respect to factors
I.
Introduction
The works by Cobb and Douglas (1928) and Kaldor (1961) created a paradigm for
macroeconomics. The idea that labor income share does not decrease or increase with time or with
the stage of development 1 had important implications in macroeconomics and growth theory.
Considering an aggregate production function, if factor income shares are constant and the price of
each factor is determined by its marginal productivity then the elasticity of output with respect to
each factor is also constant. In other words the constancy of factor shares implies that the CobbDouglas is a good approximation for the aggregate production function.
Subscribing to this paradigm, almost all of the literature on economic growth accounting assumes
that the elasticity of output with respect to capital (and labor) is constant and have concluded that
the major part of economic growth is not explained by factor accumulation but by growth in TFP
1
Recently the result was confirmed by Gollin (2002) in a serious empirical work.
(see Denison, 1967; Easterly and Levine, 2002; Solow, 1957 or Young, 1994 among others).
However there are 3 theoretical reasons why the elasticity of output with respect to reproducible
factors, namely, physical capital and human capital, should be positively correlated with the stage
of development:
1. International Trade. Eli Hecksher (1919) and Bertil Ohlin (1939) argue that comparative
advantage arises from differences in national factor endowments. The Hecksher-Ohlin theory
predicts that countries specialize and export those goods that make intensive use of locally
abundant factors, while importing (producing less) goods that make intensive use of factors that
are locally scarce. In other words, countries specialize in goods characterized by a high
elasticity of output with respect to locally abundant factors. Therefore, international trade and
specialization should increase the elasticity of aggregate output with respect to abundant factors
and decrease the elasticity of aggregate output with respect to scarce factors. Finally, if factor
income shares are equal to the elasticity of aggregate output with respect to factors then
international trade and specialization should increase the income share of abundant factors and
decrease the income share of scarce factors.
2. Foreign Direct Investment (FDI). According to John Dunning (1988) location-specific
advantages can help explain the nature and direction of FDI. By location specific advantages,
Dunning means the advantages that arise from using resource and factor endowments that are
tied to a particular location. Therefore, firms producing labor-intensive goods are likely to
invest in labor abundant countries while firms producing capital- intensive goods are likely to
invest in capital abundant countries. This implies that the behavior of FDI generates, in each
country, an expansion in the sectors that use intensively abundant factors and, as a result,
should increase the elasticity of aggregate output with respect to abundant factors and the
income share of the abundant factors.
3. Factor Saving Innovations. Economic growth models of biased innovations have been
proposed by Kennedy (1964), Zeira (1998), Acemoglu (2002), Boldrin and Levine (2002) and
Zuleta (2003) among others. In these models factor scarcity generates incentives to invest in
factor saving innovations, that is, people invest to reduce the need of scarce factors and increase
the relative use of abundant factors. If factor prices are determined by marginal productivity of
factors then laborsaving innovations reduce the income share of workers and increase capital
income share. In more general terms, the income share of non-reproducible factors decreases
with the stage of development while the income share of reproducible factors increases (Zuleta,
2003).
If factor prices are determined by their marginal productivity and, consequently, the income share
of each factor is identical to the elasticity of output with respect to such a factor, then the three
theoretical arguments described above imply that the income share of reproducible factors should
be positively correlated with the stage of development. So how can we explain the absence of a
trend for capital income share? In the following lines we propose two complementary explanations:
1. Labor Intensive Sector.
In economic growth, theorists are used to working with aggregate production functions.
However, different sectors are likely to have different production functions and reveal different
behaviors. In particular, consider a very labor intensive non-tradable sector where the
possibilities of labor-saving innovations are low, namely, services, and suppose that this sector
produces a normal good (X). The other sector produces the good Y, which is tradable and uses
capital in a more intensive way. Under these circumstances, as an economy grows the demand
for the good X also grows, which creates the need to hire more workers. As capital grows the
cost of labor also increases, and so does the relative price of good X.
If the effect of the increase in prices (in sector X) on labor demand is higher than the effect of
the increase in capital (Y), then, as the economy grows, more labor is allocated to the production
of good X. Under such circumstances labor income share decreases as the stock of capital
grows. As it will become clear, this happens when the elasticity of substitution between goods X
and Y is low enough. If this is the case, the question to ask is why capital income share does not
decrease with the stage of development. The answer to this question can be found in the
International Trade, Foreign Direct Investment and Factor Saving Innovations theories.
2. More than 2 factors.
The standard measures of labor income share includes raw labor and human capital income
shares. In the same way, the standard measure of capital income share includes land income
share. Therefore, it is possible that an increasing trend in physical capital income share is
compensated by a decreasing trend in land income share. Similarly, an increasing trend in
human capital income share may be compensated by a decreasing trend in raw labor income
share.
The rest of this paper is organized as follows. In Section 2 we present a two-sector model where
labor income share increases with the stage of development if aggregate technology is constant. In
Section 3 we present an empirical exercise; using the standard growth regression methodology, we
take into account the fact that factor shares may depend on factor abundance and verify if there
exists a constant relation between elasticity of output with respect to reproducible factors and
income per capita. Section 4 concludes.
II.
Two Sector Model
Consider a two-sector model, where each sector produces a different good. Both goods X and Y are
normal goods and their production functions are as follows:
(1) Y = K α L y
1−α
X = BL x
where K is capital, L is labor, the sub-indexes x and y account for the amount of labor devoted to the
production of goods X and Y respectively, B is the labor productivity in the production of the good
X, α is the elasticity of output with respect to capital in the production of the good Y, and 1-α is the
elasticity of output with respect to labor in the production of the good Y.
For simplicity we assume constant labor supply and normalize L=1. Therefore, Ly=1-Lx.
We also call k the capital labor ratio in sector Y, namely, k =
K
; use good Y as a
1 − Lx
numeraire, and call p the relative price of good X. Markets are competitive so wage (w) is
determined by the marginal productivity of labor, and the interest rate (r) is determined by
the marginal productivity of capital, namely,
(2) w = (1 − α )k α = pB and r = αk α −1 .
Given that both goods are normal we can assume a CES utility function,
(
U = Yφ + X φ
)
1
φ
where the elasticity of substitution between goods X and Y is given by, ε X ,Y = 1−1φ .
Therefore, if the elastic ity of substitution is positive, then φ≤1, and if it is lower than one
then φ≤0. If agents are utility maximizers then relative prices and consumption should relate in the
following way:
 C
p =  Y
C X



1− φ
where Cy and Cx are the consumption of good Y and X respectively. Therefore if the relative price
of good X grows then consumers substitute consumption of good X for consumption of good Y.
For simplicity let me assume that total demand behaves in the same way that consumption does,
I
that is, the demand for investment (I) of goods X and Y is such that p =  Y
IX
1−φ
Y
(4) p =  
X
 K α (1 − L) 1−α
= 
BL




1− φ
. In this case,
1−φ



Using equation 2 and rearranging yields
1


(
1 − α )α K 
 1
(4´) p = 
1
 1 − α pB − (1 − α )α K 


1 −φ
φ
From equation 4´ it follows that in the absence of labor saving innovations (constant α) the price of
good X is positively correlated with the stock of capital K. Combining equations 4 and 2 we get:
1
 B  αφ
(5) K = 
 (1 − L x )
 (1 − α ) 
From equation 5, if φ >
1−φ (1 −α )
αφ
φ −1
(L x ) αφ .
2
or φ < 0 then ∂ K > 0 . In other words, if the substitutability
2 −α
∂L x
between X and Y is low enough, namely, ε X ,Y < ( 2−αα ) , then as the economy accumulates capital,
the share of workers devoting their time to the production of good X grows2 . Therefore: (i) given
the elasticity of output with respect to capital (α), for low elasticity of substitution between capital
and labor there exists a positive relation between capital and share of labor in the production of
good X. Moreover, if ε X ,Y < 1 then ∂ K > 0 for any α<1. And (ii) given the elasticity of
∂L x
2
If 0 < φ <
2 then 1 < ε and 2 − α
<ε
2−α
α
substitution between capital and labor, for low levels of elasticity of output with respect to capital
(α) there exists a positive relation between capital and share of labor in the production of good X ,
that is, ∂ K > 0 3 . Therefore, for reasonable values of ε X ,Y and α there exists a positive relation
∂L x
between the stock of capital and the share of workers in the production of good X4 .
Now, the labor income share of the economy (LISh) is given by the ratio between wage (pB)
multiplied by the number of workers (L=1) and total output ( AK α L y
LISh =
1−α
+ pBL x ):
pB
K (1 − Lx )1−α + pBL x
α
Using equation 2 and rearranging,
(6) LISh =
1− α
2 − L x (1 + α )
From equation 6 it follows that if the elasticity of output with respect to capital (α) is constant then
labor income share is positively correlated with the share of workers devoted to the production of
the good X.
Finally, from equations 5 and 6 it follows that, if ε X ,Y <
2 −α
α
, then labor income share grows as K
grows, that is, as the economies grow labor income share grows. This result has the following
implications: (i) if ε X ,Y <
2 −α
α
and the elasticity of output with respect to capital is constant, then as
the economy grows labor income share grows; and (ii) if the elasticity of output with respect to
capital is low, then as the economy grows, labor income share grows.
From table 1, it is evident that for all the countries in the sample there exists a positive trend in the
share of workers devoted to the production of services. Therefore, if the elasticity of output with
respect to capital is constant, then labor income share should have a positive trend.
3
4
Note that if ε=2 then α<0.67 implies ∂ K > 0 , and if α=0.4 then ε<4 implies ∂ K > 0 .
∂L x
∂L x
Estimates of Armington elasticities, that is , the elasticity of substitution between home and imported goods,
have been supplied by a variety of scholars (see Blonigen and Wilson, 1999, for example). In these studies it
is found that the Armington elasticities generally fall below 1.5. Given this result it is likely that the elasticity
of substitution between tradable and non tradable goods is below 1.5.
Table 1
Italy
Spain
Australia
Japan
Germany
Finland
Switzerland
France
Sweden
Austria
United Kingdom
Norway
Canada
USA
Source: OECD
III.
Employment in Services as a percentage of civilian employment
1965
1970
1975
1980
1985
1990
36.8
40.3
44.1
47.8
55.2
58.5
39.7
44.9
50.1
54.9
38
40.5
46.6
49.3
52.9
55.3
44.7
46.9
51.5
54.2
56.4
58.7
40.7
42.9
47.6
51
53
57.3
51.8
56.5
60.7
41.2
45.4
55
58.3
63.6
43.1
47.2
51.1
55.5
60.8
64.6
62.1
66.1
67.8
53.1
55
59.4
62.5
66.3
69.2
49.5
52
59.8
65.8
68.7
56.1
61.6
65.4
68.8
62.6
64.3
66.9
70.5
71.7
61.1
65.4
65.9
68.8
70.9
1995
59.2
61
60.3
60.7
60.5
64.5
66.9
69
71
72.1
70.1
71.4
74
73.1
Physical and Human Capital
To test the hypothesis that increases in physical capital (human capital) income share is
compensated by decreases in land (raw labor) income share, we proceed as follows: First,
in panel data regression we estimate the relevant parameters related to the elasticity of
output with respect to land, physical capital and human capital. Second, by using the
estimated parameters, we calculate the elasticity of output with respect to the factors.
The existing literature of empirical economic growth has two main branches. Authors like
Barro and Sala- i-Martin (1992) and Mankiw, Romer and Weil (1992) assume that factor
shares are equal across countries. Others, such as Durlauf and Johnson (1995) allow for
differences in factor shares between different groups of countries but assume that factor
shares are the same within groups. We need to go one step ahead and allow for differences
in factor shares not only among groups, but also among countries within groups. To do so,
we use the standard growth regression methodology taking into account the fact that factor
shares may depend on factor abundance.
2000
62.2
62.3
63.6
63.7
63.9
66.3
69.8
72
72.9
73.1
73.2
73.8
74.1
74.4
To test the model, we use the data of De la Fuente and Domenech (2001) from OECD5
countries for GDP, physical capital and human capital. For the variables land (T) and raw
labor (L) we use data from the World Development Indicators.
Consider an aggregate production function, which combines two reproducible factors,
human and physical capital (H and K) and two non-reproducible factors, land and raw labor
(T and L) in a Cobb-Douglas: Y = AK α H β T γ L1−α − β −λ . Therefore, income per capital is
y = Ak α h β t γ , where k is capital per worker, h is human capital per worker and t is land per
worker. By taking logarithms and differences between period zero and period one, we obtain:
log( y1 ) − log( y0 ) = log( A1 ) − log( A0 ) +α1 log(k1) −α 0 log(k0 ) + β1 log(h1) − β0 log(h0 ) +γ 1 log(t1) − γ 0 log(t0 )
We assume that the elasticity of output with respect to the different factors can be different at the
beginning of different periods. However, since we are considering periods of five years we assume
that the effect of the increments in the elasticities are negligible. Therefore, the growth rate of
income per capita is given by:
∆y
∆A
=
+α
y
A
0
∆k
+ β
k
0
∆h
+ γ
h
0
∆t
t
For the growth rate of total factor productivity we test for technological convergence. For each
country i, at any time, TFP can be obtained as Ai =
yi
. Therefore, if there is technological
t i kiα i hiαi
γi
convergence, then
∆Ai
= µ + δ (log y 0 ,i − α 0, i log k 0, i − β 0 ,i log h0 ,i − γ 0, i log t 0, i ) + ε i
Ai
where the parameter δ indicates the speed of technological convergence, ε is a change in
technology that is not explained by technological convergence and is country specific and ε i is
partially captured by country and time dummies.
5
Countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy,
Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, USA, UK. Years: 1960,
1965, 1970, 1975, 1980, 1985, 1990, 1995.
The changes in α are reflected in a reduction of γ, so γ = ρ − α , where ρ is a parameter to be
estimated. For the functions a and ß we use the functions α=η(1+ log(k)/( log(k)+1)) and β=λ(1+
log(h)/( log(h)+1)), respectively 6 . In this way, the elasticity of output with respect to reproducible
factors has two components, one is fixed, and the other one is variable and bounded from above.
Finally, we use two different measures for the variable T: the first is the total area of the country
measured in hectares, and the second is the arable land measured in hectares.
Table 2 presents the results. The coefficients γ, λ and ρ are positive and significantly different from
zero in both cases. Thus evidence supports the idea that the contribution of physical and human
capital accumulation to economic growth is higher in capital abundant economies. Additionally,
a+γ is 0.35 in one case and 0.36 in the other. These numbers are similar to the conventional wisdom
which estimates capital income share around 1/3.
Table 2
1
2
Total Area
Arable Land
η
0.198
(0.000)
0.186
(0.000)
λ
0.325
(0.000)
0.377
(0.000)
ρ
0.360
0.349
(0.000)
(0.000)
0.074
(0.000)
0.094
(0.000)
0.789
123
0.092
(0.000)
0.133
(0.000)
0.650
100
δ
µ
Adjusted R-square
Number of Observations
Country Dummies: Belgium, USA, Canada, France, Italy, Netherlands, and Spain.
Time Dummies: 1970, 1975, and 1980
Multiplicative Dummy for change in physical capital: Ireland
Multiplicative Dummy for change in human capital: Finland, Ireland and New Zealand.
6
Other specifications were considered and the results didn’t change.
Finally, we use the data and the estimated parameters to calculate the values of α, β and ρ for all the
countries in different years. Figure 1 shows that, according to our, results there exists a positive
correlation between the elasticity of output with respect to reproducible factors and the level of
output per worker.
Elasticity of Output with Respect to Factors
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Physical Capital
Human Capital
Land
Raw Labor
0.05
0
1.80
2.30
2.80
3.30
3.80
4.30
log GDP per capita
IV.
Conclusions
International Trade, Foreign Direct Investment and Factor Saving Innovations theories provide
three different theoretical reasons why the elasticity of output with respect to reproducible factors,
namely, physical capital and human capital should be positively correlated with the stage of
development. Similarly, if factor prices are determined by factor marginal productivities then the
elasticity of output with respect to factors determines factor income shares. This implies that there
are 3 different theoretical reasons why the income share of reproducible factors should be positively
correlated with the stage of development
In this paper we present two reasons why labor income share can be constant even if factor prices
are determined by factor marginal productivities and the elasticity of output with respect to
reproducible factors increases with the stage of development. The first reason is the existence of a
labor-intensive sector that is not subject to labor saving innovations. The second reason is the
existence of more than one reproducible factor and more than one non-reproducible factor.
The main point of the paper is that, even though labor income share does not present any trend, the
elasticity of output with respect to reproducible factors depends on the stage of development of the
countries. This result has important implications for empirical exercises related to economic growth
(accounting and econometrics). In particular, the conclusion that the major part of economic
growth is not explained by factor accumulation but by growth in TFP needs to be revised.
Similarly, the endogeneity of the elasticity of output with respect to reproducible factors has
implications for fiscal policy.
Bibliography
1. Acemoglu, Daron "Directed Technical Change" the Review of Economic Studies, October
2002, 69 (4) .
2. Boldrin, Michele and Levine, David K. " Factor Saving Innovation" Journal of Economic
Theory, July 2002, 105 (1), 18-41
3. Cobb, Charles and Douglas, Paul H, “A theory of production”, American Economic Review
1928, 18 , 139–165
4. Blonigen, Bruce and Wilson, Wesley “Explaining Armington: What Determines
Substitutability between Home and Foreing Goods” Canadian Journal of Economics 1999,
32 (1), 1-21
5. Denison (1962) Sources of Economic Growth in the United States and the Alternatives
Before Us, New York: Committee for Economic Development.
6. Dunning, J.H. Explaining International Production London: Unwin Hyman, 1988
7. Gollin, Douglas "Getting Income Shares Right" Journal of Political Economy April 2002,
110 (2), 458-474.
8. Heckscher, Eli F. “The Effect of Foreign Trade on the Distribution of Income", 1919,
Ekonomisk Tidskrift
9. Kaldor, Nicholas. 1961. “Capital Accumulation and Economic Growth.” In FA
Lutz and DC Hague, eds., The Theory of Capital, 177-222. New York St, Martin’s Press.
10. Kennedy, Charles "Induced Bias in Innovation and the Theory of Distribution", Economic
Journal, LXXIV 1964, 541-547.
11. Ohlin, B Interregional and International Trade. Cambridge: Harvard University Press,
1933.
12. Solow, Robert (1957) “Technical Change and the Aggregate Production Function” Revie w
of Economics and Statistics, 39, 312-320.
13. Young, Alwyn (1995) The Tyranny of Numbers: Confronting the Statistical Realities of the
East Asian Growth Experience” Quarterly Journal of Economics, 110: 641-680
14. Zeira, Joseph, "Workers, Machines and Economic Growth" Quarterly Journal of
Economics, 1998, 113 (4), 1091-1118.
15. Zuleta, Hernando (2003) Essays on Factor Saving Innovations PhD Dissertation Universitat
Pompeu Fabra.